Refrigerators heretofore known include a heat exchanger serving as a cooler and disposed within a storage compartment, such that the air in the storage compartment is cooled by the cooler in direct contact therewith. Simultaneously with refrigeration, therefore, the water in the air within the refrigerator freezes on the surface of the cooler and covers the cooler with frost, consequently hampering the continuous operation of the cooler and drying the refrigerated foods due to the removal of water. For defrosting, the cooler must be heated suitably from time to time, but this gives rise to other attendant problems such as marked heat losses. Japanese Utility Model Publication for Objection No. 2119/1975, published on Jan. 21, 1975, discloses an improved refrigerator including a closed vegetable box placed in a storage compartment for preventing vegetables from drying, a cooler disposed in a freezer compartment above the storage compartment, and a circulation passage provided between the storage compartment and the freezer compartment for passing cold air along the wall of the closed box. With this structure, cold air flows out from the freezer compartment into the storage compartment, refrigerating the closed box from outside, and then returns to the freezer compartment in which the air comes into contact with the cooler. Accordingly, the refrigerator still involves problems in that the cold air comes into direct contact with the foods stored in the freezer and storage chambers, drying the foods by absorption of water and depositing frost on the surface of the cooler.